Concept explainers
Why the acceleration of a freely falling body is less in the Galapagos island at the equator and more a Oslo, Norway?
Answer to Problem 7C
The acceleration of free fall is different at different latitudes because of the following reasons:
(i) The radius of the earth is greatest at the equator and the acceleration of free fall at points close to the surface of the earth is inversely proportional to the square of the radius of earth at the point.
(ii) Due to rotation of the earth, the points on the surface of the earth experience an outward force depending on the latitude.
Explanation of Solution
Introduction:
Acceleration of free fall is the acceleration experienced by a body due to the force of gravitation acting on it. This is given by the expression,
Here, G is the universal gravitational constant, M is the mass of the Earth and R is the radius of the earth.
(i) Earth is not a perfect sphere. It is bulged at the equator and flattened at the poles. The equatorial radius is estimated to be 6378 km and the polar radius is around 6357 km. The difference between the equatorial and polar radius is about 21 km.
Since
(ii) The earth rotates from west to east with an
Consider a point A located at a latitude
Resolve the gravitational force mg into two components,
The net force along AB is given by,
From the triangle OBA,
Therefore,
The force along AX is given by,
The net force acting on the mass is given by,
From Newtons second law,
Here,
From equations (1),(2), (3) and (4),
Simplify the expression.
The value of
Hence,
Using binomial expansion and neglecting all higher powers of
At equator the latitude is zero,
At the poles, the latitude is 90°, the acceleration of free fall at poles is given by,
It can be seen that the value of
Conclusion:
Thus, the acceleration of free fall is different at different latitudes because of the following reasons:
(i) The radius of the earth is greatest at the equator and the acceleration of free fall at points close to the surface of the earth is inversely proportional to the square of the radius of earth at the point.
(ii) Due to rotation of the earth, the points on the surface of the earth experience an outward force depending on the latitude.
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